I wonder if there is analytical way to derive all branch values for W(-pi/2) as function of k and hence h(e^pi/2). What does Mathematica says about the values of h(e^pi/2) and h(e^-pi/2) along few first branches? (I have no access to such instruments...). Any kind of pattern?

From paper I have I see these formulas for Quadratix of Hippias but can not yet understand how these could be used to derive values of W(-pi/2) on other branches than 0 and -1.

I still think there are actually 4 values h(e^pi/2) and may be even in some more general divergent cases of real number tetration: 2 corresponding to -i, 2 to +i , but I do not know yet how to find out if that is true and what are these values. The only case when there are only 2 distinquishable values ar +-i bacause the real parts here are 0..

I also make a conjecture that as number of branch k-> infinity,
h(e^pi/2) = -1 independent of way it is reached.

Also, I think there must be some relation between all h(e^pi/2) and
h(e^-pi/2) branch values, possibly type of Mobius transform. Again , just a conjecture, to be proved wrong to make progress

nice coincidence. I found a formula in this paper, which agrees perfectly with my fixpoints-for-real-bases b>e^(1/e) formula. In that article they discuss boundaries for the lambert-w-function such that
x = eta ctg(eta) + i * eta
and similar (page 15, formula 4.1 - 4.5)

For the complex fixpoints for real b>e^(1/e) I had the same type of formula; such that
u = beta cos(beta)/sin(beta) + I * beta
t = exp(u)
b = exp(u/t) is real
and this last expression is a branch-enabled version of
b = t^(1/t)
Nice...

Gottfried Wrote:For the complex fixpoints for real b>e^(1/e) I had the same type of formula; such that
u = beta cos(beta)/sin(beta) + I * beta
t = exp(u)
b = exp(u/t) is real
and this last expression is a branch-enabled version of
b = t^(1/t)

Quote:So how do you finally compute the th branch of or with this formula?

Yes, how would You? And particularly, in case

h(e^pi/2) and h(e^-pi/2)?

Waiting impatiently even

Ivars

No chance.... ;-)

Yes, there is the coincidence; but I did not compute the W-function but the function, which gives real bases b (or, to avoid confusion with the parameter-notation in the article:bases s) for exp(u/t) where s=t^(1/t) is the principal branch, given u (or more precisely: given the imaginary part of u), and get real bases s>1 .

I set imag(u) = beta = any real value -pi < beta < pi
compute real(u) according to the above formula thus having u completed,
then compute t = exp(u)
and then compute s = exp(u/t), which is then surely real.

The formula for this is in the file fixpoints.pdf
In this article I've not yet included the extension of the range for beta>pi; but I've done some computations with this and found further fixpoints for the real s in the regions 2*k*pi< beta < 2*(k+1)*pi. However, there is no exact periodicity, the consecutive fixpoints for the same s approach the lower bound 2*k*pi with the index k.

I can find the inverse, to compute a fixpoint t by a given s, only by numerical approximation, since s=f(u) is monotonic, using a binary iterative process.
If I *had* the branch-enabled Lambert-W-function, this would be easier, but Pari/GP doesn't have it and I still have only the python-example from wikipedia for the real-valued region, and have not yet invested in programming it myself.

It is not that difficult to compute the branches of .
The branches of are simply the fixed points of . In this way you can compute the branches via , where .
This is however valid only for as the numbering of the branches in this interval may be a bit different. For the branch 0 and -1 (which are non-real) join and for slightly smaller it splits again into two real branches.
The negative branches are simply the conjugates of the positive branches:.

Edit:
As it is so easy to compute it is perhaps more appropriate to compute from by instead of computing from .

Ivars Wrote:PLEASE post at least one numerical example how to use this limit formula e.f for k=-2, h(e^(pi/2)).

I am not familiar (entirely my fault) with the notations You use - what is n, how to take the limit, what is z, what is b (I assume b=e^(pi/2) in my case.

Yes the notation, mainly you have to know what means, it is simply the times repetition of applied to , for example .

In our example is then . Because we compute which means the complex conjugate of . To compute this we consider and then compute . This can of course only approximately computed. Have a look at some sample values: